Subgroup ($H$) information
| Description: | $C_2^2\times C_{14}$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Index: | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$\langle(1,2)(3,5)(4,6)(7,10,14,8,15,9,13), (1,2), (3,5)(4,6), (3,6)(4,5)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $D_4\times \SL(2,8):C_6$ |
| Order: | \(24192\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\SL(2,8).C_3\times C_2\wr C_2^2$, of order \(96768\)\(\medspace = 2^{9} \cdot 3^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_6\times \GL(3,2)$, of order \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
| $W$ | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $72$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $2$ |
| Projective image | $C_2^2\times {}^2G(2,3)$ |